《结构化学 Structural Chemistry》课程教学资源（课件讲稿）Chapter 2 Atomic Structure

Dalton(1803)History ofThomson(1904)(positive andnegativecharges)Atomic ModelsRutherford (1911)(the.nucleus).Understanding atomicstructure is the first step tounderstandtheStructuresofBohr(1913)(energylevels)Matters·The so-called electrondensityis actuallytheprobabilitydensityofelectronwave!Schrodinger(1926)(electroncloudmodel)12Ouantummechanics

History of Atomic Models •Understanding atomic structure is the first step to understand the Structures of Matters. •The so-called electron density is actually the probability density of electron wave! Quantum mechanics

2.1 The Schridinger equation anditssolution for one-electron atomsHatom, Het and Li2+2.1.1The Schrodinger equationThe Hamiltonian of a one-electron atom/cation.h?h?72+V72H=T,+T +Vn-enen-e8元8元'mmnnucleuselectronThe Hamiltonianfor amany-electron atom? For a many-electron atom, the kinetic-energyoperator should sum up the contribution from everyelectron.The potential energyfunction should12include all those from n-e and e-e interactions

2.1 The Schrödinger equation and its solution for one-electron atoms 2.1.1 The Schrödinger equation • The Hamiltonian of a one-electron atom/cation, H atom, He+ and Li2+ e n e e n n n e n e V m h m h H  T T V         ˆ 8 8 ˆ ˆ ˆ ˆ 2 2 2 2 2 2   N e • The Hamiltonian for a many-electron atom? • For a many-electron atom, the kinetic-energy operator should sum up the contribution from every electron. The potential energy function should include all those from n-e and e-e interactions. nucleus electron

Note that the nucleus is much heavier than the electron and theelectron moves much quickly around the atomic nucleus! TheHamiltonian can be simplified ashA=T.+VOne-electronHamiltoniann-e8元mhaL+n-e8元2maxZe?+ 7n-e4元Hy=EyThe SchrodingerequationSeparationofvariables (x,y,z)hinderedbyr?!12

e n e e e n e V m h H  T V       ˆ 8 ˆ ˆ ˆ 2 2 2  r Ze Vn e 0 2 4 ˆ      Note that the nucleus is much heavier than the electron and the electron moves much quickly around the atomic nucleus! The Hamiltonian can be simplified as r e 2 2 2 r  x  y  z n-e e ) V x y z ( π m h H ˆ 8 ˆ 2 2 2 2 2 2 2 2            Separation of variables (x,y,z) hindered by r ?! The Schrödinger equation H ˆ   E One-electron Hamiltonian

SphericalpolarcoordinatesZe(r0s)<r<+80x=rsinocos0≤0≤元y=rsinesingz=rcoso0≤≤2元+z/xZcosO:r : distance from origin (nucleus)+1/xO : angle drop from the z-axis.tgb = y/ x@ : angle from the x-axis (on the x-y plane)(xy,z) (r,e,);Y(x,y,z) = (r, e, Φ)12Is it possible to make Y(r, e, Φ) =R@(OΦ@)

Spherical polar coordinates 2 2 2 r  x  y  z (x,y,z)  (r,q,f); (x,y,z)  (r, q, f) r : distance from origin (nucleus). q : angle drop from the z-axis. f : angle from the x-axis (on the x-y plane) 2 2 2 cos x y z z   q  tgf  y / x 0 < r < + 0  q   0  f  2 Is it possible to make (r, q, f)  R(r)(q)F(f) ?

Using spherical polar coordinates, we haveaa11aasinHarar00A002insinh?ZeAn-e8元°m(4元)rThe SchrodingerequationH(r.0.b)= Ew(r,.dbecomesaC?1a1Ze(sin y=EyorOr008元a0sinHad4元sinmh2 /(8元2mdivided bya118元Te0am21=0Esinh?orsingaga0sin?OrAad4元8012Thus it is reasonable to suppose y(r, e, Φ) = R(r)@(0)Φ(Φ)

Using spherical polar coordinates, we have 2 2 2 2 2 2 2 2 sin 1 (sin ) sin 1 ( ) 1 q q f q q q               r r r r r r 2 0 (4 ) n e Ze V  r    2 2 2 2 2 2 2 2 2 2 0 1 1 1 8 ( ) (sin ) ( ) 0 sin sin 4 me Ze r E r r r r r h r     q  q q q q f                 The Schrödinger equation ( , , ) ( , , ) becomes ˆ H r q f  E r q f divided by n-e e V π m h H ˆ 8 ˆ 2 2 2       q q f   q  q q E r Ze r r r r m r r h e                           0 2 2 2 2 2 2 2 2 2 2 sin 4 1 (sin ) sin 1 ( ) 1 8 /(8 ) 2 2  h  me Thus it is reasonable to suppose (r, q, f)  R(r)(q)F(f)